Friendship ties are essential constructs in on-line social networks, as witnessed by friendship between Facebook users or followers on Twitter. Another pervasive aspect of social networks are negative ties, where users may be viewed as adversaries, competitors, or enemies. For further background on ties in social networks and more generally, complex networks, see [5, 13, 17]. Negative ties are often hidden, but may have a powerful influence on the social network. An early example of how negative ties influences networks structure comes from the famous Zachary Karate network, where an adversarial relationship assisted in the formation of two distinct communities .
Complex networks, including on-line social networks, contain numerous mechanisms governing edge formation. In the literature, models for complex networks have exploited principles of preferential attachment [2, 4], copying or duplication [9, 10, 16], or geometric settings [1, 7, 11, 20, 23, 28]. The majority of complex network models are premised on the formation of edges via positive ties. Structural balance theory in social network analysis cites several mechanisms to complete triads, or triples of vertices. Vertices may have positive or negative ties, and triads are balanced if the signed product of their ties is positive. Hence, balanced triads are those consisting of all friends, or two enemies and a friend. These triads reflect the folkloric adages “friends of friends are friends” and “enemies of enemies are more likely friends,” respectively. Such triad closure is suggestive of an analysis of adversarial relationships between vertices as another model for edge formation. For an example, consider market graphs, the vertices are stocks, and stocks are adjacent as a function of their correlation measured by a threshold value Market graphs were considered in the case of negatively correlated (or competing) stocks, where stocks are adjacent if for some positive ; see . In social networks, negative correlation corresponds to enmity or rivalry between agents. We may also consider opposing networks formed by nation states or rival organizations, or alliances formed by mutually shared adversaries as in the game show Survivor [6, 19].
Transitivity is a pervasive and folkloric notion in social networks, and postulates triads with all positive signs. A simplified, deterministic model for transitivity was posed in [8, 9], where vertices are added over time, and for each vertex , there is a clone that is adjacent to and all of its neighbors. The resulting Iterated Local Transitivity (or ILT) model, while elementary to define, simulates many properties of social and other complex networks. For example, as shown in , graphs generated by the model densify over time, and exhibit bad spectral expansion. In addition, the ILT model generates graphs with the small world property, which requires the graphs to have low diameter and dense neighbor sets. For further properties of the ILT model, see [12, 25].
Adversarial relationships may be modeled by non-adjacency, and so we have the resulting closure of the triad as described in Figure 1.
A simplified, deterministic model simulating anti-transitivity in complex networks was introduced in . The Iterated Local Anti-Transitivity (or ILAT) model duplicates vertices in each time-step by forming anti-clone vertices. The anti-clone of is adjacent to the non-neighbor set of . Perhaps unexpectedly, graphs generated by ILAT model exhibit many properties of complex networks such as densification, small world properties, and bad spectral expansion.
In the present paper, we consider a new model synthesizing both the ILT and ILAT models, allowing for both transitive and anti-transitive steps over time. We refer to this model as the Iterated Local Model (ILM), and define it precisely in the next subsection. Informally, in ILM we are given as input an infinite binary sequence. For each positive entry in the sequence, we take a transitive, ILT-type step. Otherwise, we take an anti-transitive, ILAT-type step. Hence, ILM contains both the ILT and ILAT model as special cases, but includes infinitely many (in fact, uncountably many) other model variants as a function of the infinite binary sequence.
We consider only finite, simple, undirected graphs throughout the paper. For a graph with vertex , define the neighbor set of , written , to be The closed neighbor set of , written is the set Given a graph , we denote its complement by When it is clear from context, we suppress the subscript . For background on graph theory, the reader is directed to . Additional background on complex networks may be found in the book .
1.1. The Iterated Local Model
We now precisely define the iterated local model (ILM). First, we must define two iterative procedures on a graph by considering steps that are locally transitive or locally anti-transitive.
We define a graph as follows. For each , add a new vertex to the vertex set of such that is adjacent to all neighbors of in In particular, . The vertex is called the transitive clone of , and the resulting graph is called the graph obtained from by applying one locally transitive step.
Analogously, we define the graph as follows. For each , add a new vertex that is adjacent to all non-neighbors of in In particular, . The vertex is called the anti-transitive clone of , and the resulting graph is called the graph obtained from by applying one locally anti-transitive step.
Note that the ILT model is defined precisely by applying iterative locally transitive steps; an analogous statement holds for the ILAT model. Observe also that clones or anti-clones introduced in the same time-step are pairwise non-adjacent.
Fix an infinite binary sequence , which we refer to as an input sequence. The Iterated Local Model(ILM) graph, denoted is defined recursively. In the case we have that . For , we have that
Hence, an instance of 1 in the sequence results in a transitive step; otherwise, an anti-transitive step is taken. If the sequence contains only 1’s, then the resulting graph at each time-step is isomorphic to the graph from the same time-step of the ILT model, thus, we write . Similarly, if the sequence contains only 0’s, the resulting graph is isomorphic to the graph from the same time-step of the ILAT model and we write . We use the simplified notation ILM graph for a graph for any choice of , , or . ILT and ILAT graphs are defined in an analogous fashion.
We will denote the order of the ILM graph as and denote the size (that is, number of edges) by The (open) neighborhood of a vertex at time by
and the closed neighborhood by The degree of a vertex is written as To simplify this notation, when the sequence and initial graph are clear from context, we will simply write , , and for the order, the size, the neighborhoods, and the degree of a vertex at time , respectively. We emphasize that is an induced subgraph of , and as such we will always consider to be embedded in in the natural way. That is, expressions of the form may be used for vertices to refer to the degree in the induced subgraph isomorphic to .
In the present paper, we analyze various properties of ILM graphs. While the ILT and ILAT graphs satisfy various properties such as low diameter and densification, those statements are not a priori obvious for the ILM model. We prove in Theorem 3 that ILM generates graphs that densify over time. We do this by deriving the asymptotic size of ILM graphs given by the following expression:
where is the largest index less than such that . The clustering coefficient of ILM graphs is studied in Section 3. We say has bounded gaps between ’s, or simply bounded gaps if there exists some constant such that there is no string of contiguous ’s. In contrast to the known clustering results for the ILT model in , we will see that for any sequence with bounded gaps, the clustering coefficient is bounded away from . Our results here are also the first rigorously presented results for the clustering coefficient of the ILAT model ; further, our results give an improvement on bounds for the clustering coefficient of ILT graphs.
Graph theoretical properties of ILM graphs are of interest in their own right. In Section 4, we explore classical graph parameters, including the chromatic and domination numbers, and diameter. The domination number of ILM graphs is eventually either 2 or 3, and Theorem 17classifies exactly when each value occurs. The diameter of an ILM graph eventually becomes , usually after only two anti-transitive steps; see Theorem 20. Bad spectral expansion for the ILM graphs is proven in Section 5. Section 6 includes a discussion of the Hamiltonicity of ILM graphs, proving that eventually, all ILM graphs are Hamiltonian. In addition, ILM graphs eventually contain isomorphic copies of any fixed finite graph. Hamiltonicity and induced subgraph properties were not perviously investigated in the ILT or ILAT models. We finish with a section of open problems and further directions.
2. Density and Densification
Complex networks often exhibit densification, where the number of edges grows faster than the number of vertices; see . Both the ILT  and ILAT models  generate sequences of graphs whose edges grow super-linearly in the number of vertices. We now show a number of results regarding the number of edges in an ILM graph. The following theorem from  gives us the average degree of an ILT graph, which will be useful for studying ILM graphs. We define the volume of a graph by
Note that the average degree of equals
 Let be a graph. For all integers , the average degree of equals
The analogous theorem for ILAT graphs is the following.
 Let be a graph. For all integers , the average degree of equals
We now provide an asymptotic formula for the number of edges in an ILM graph.
Let be a binary sequence with at least one , let be the least index such that , and let . Let be the largest index less than or equal to such that . For any graph and ,
Let . Suppose in the first case that (that is, when ). We then have that for all , , since for all vertices , is a non-neighbor of if and only if the clone of is adjacent to . Therefore, we have that
We then have that
and this case is finished.
Thus, we have that
As a corollary to Theorem 3, we have that ILM graphs also undergo densification for any input sequence .
For any binary sequence and initial graph , we have that
We may also consider sequences with bounded gaps between ’s. Results in this case follows immediately from the fact that for such sequences, .
If is a binary sequence with bounded gaps between 0’s, then for any graph , , where the implied constant depends on the size of the largest gap in .
When given a specific input sequence, we can use these recurrences to say much more about the number of edges. Here, we give a much stronger asymptotic result for alternating sequences.
If we consider the alternating sequence , then for even we have that
3. Clustering Coefficient
Given a graph , the local clustering coefficient of a vertex is defined by
That is, gives a normalized count of the number of edges in the subgraph induced by the neighbor set of in . The clustering coefficient of is given by
Complex networks often exhibit high clustering, as measured by their clustering coefficients . Informally, clustering measures local density. For the ILT model, the clustering coefficient tends to as , although it does so at a slower rate than binomial random graphs with the same average degree . More precisely, we have the following.
In contrast to Theorem 7, we will see that for any sequence with bounded gaps, the clustering coefficient is bounded away from for the ILM graphs. We will find it useful to write for , and when and are clear from context, we may simply write , consistent with earlier defined notation. We establish a bound on the change in the clustering coefficient from performing a transitive step.
If is a graph with minimum degree , then
Let , and let be its transitive clone in . Let denote the local clustering coefficient of in , while will denote the local clustering coefficient of in .
Note that since , and is a dominating vertex in . Recall that and are the open and closed neighborhoods of in , in this case since we are only doing a single transitive step, we will only use these expressions for or . We will give a lower bound for in terms of .
We now count the edges in . In the induced subgraph , there are
many edges. There are
many edges between vertices in , and . Finally, there are edges incident with . This accounts for all the edges in . Since , this gives us that for all ,
Now, recall that for all , we have . Thus,
completing the proof of Lemma 8. ∎
With the preceding lemma, we can improve the lower bound on Theorem 7 to derive a slightly better bound for the clustering coefficient of ILT.
If is a graph, then
Towards bounding the clustering coefficient for sequences with bounded gaps, we show that the clustering coefficient is bounded away from whenever we perform an anti-transitive step.
Let be a graph and be a binary sequence with bounded gaps between 0’s. Let be the constant such that there is no gap of length , and let be the third index such that . For all , if , then
Let , , and let , and be the two largest indices such that , and . We then have that and .
If , then . If , then we have that
Since all entries between an are 1, we have, inductively that
Thus, is adjacent to exactly half of the vertices in .
Let and . Therefore, we find that .
If , then we define to be the vertices in and the clones of vertices in born at time . Similarly, let be the vertices in along the clones of born at time . Note that there are no edges between and and . In addition, , while .
Inductively, we can continue in this fashion until we have sets and of size with no edges between them and , while . After an anti-transitive step, the vertices in and will be adjacent to every clone of a vertex in . We then have that contains at least edges. Since , we have that
Note this is holds for all vertices . There are such vertices, and hence,
completing the proof of Lemma 10 ∎
We now have the tools to prove the main result of this section.
Let be a graph, be a binary sequence with bounded gaps between zeroes, and be an absolute constant such that there is no gap of length . If is the third index such that , then for all , the clustering coefficient
Let . Let denote the largest index such that , and so . Note that since at least one anti-transitive step happens before time , we have that the maximum degree . Thus, after the anti-transitive step at time , we have minimum degree . Note that the minimum degree does not decrease after a transitive step, so for all .
4. Graph Parameters
In this section, we will explore a number of classical graph parameters for ILM graphs, including the chromatic number, domination number, and diameter. For a more detailed discussion of graph parameters, the reader is pointed to .
A shortest path between two vertices is called a geodesic. The diameter of a graph is the length of the longest geodesic; that is, it is the furthest distance between any two vertices. The radius of a graph is the largest integer such that every vertex has at least one vertex at distance from it.
If is a graph, then . If the radius of is at least , then .
Observe first that after a transitive step or an anti-transitive step, the vertex set of resulting graph can be partitioned into a set which induces and an independent set. Thus, the chromatic number goes up by at most one.
Let and be graphs such that is an induced subgraph of . We claim that if for every vertex , there exists a vertex with , then .
Now, assume to the contrary that This induces a proper coloring on . It is well-known and straightforward to see that if is properly colored with colors, then there must be a vertex such that has a vertex of every color in it. Hence, there is no possible color for the clone , a contradiction.
Similarly, in , for every vertex , if has radius at least , there exists a vertex at distance from , so the anti-transitive clone has the property that . Thus, by the same argument in the preceding paragraph, . ∎
We prove that after one anti-transitive step, the radius is at least , and transitive steps preserve the radius.
If is any graph, then the radius of is at least three. Further, if is a graph with radius at least , then the radius of is at least 3.
To see the first part of the lemma, note that if , and is the anti-transitive clone of , then , so .
For the second part of the lemma, first, note that if , then the distance between and in is the same as the distance in . Indeed, since for any vertex , and its transitive clone , we have , no --geodesic will contain both and , and furthermore, for any --geodesic containing , we can replace with , giving us another --geodesic. We then have that there is a --geodesic using only vertices in , so the distance between and is the same in both and , so every vertex in has a vertex at distance . Note that if for , then for the transitive clone, of , we also have , proving the proposition. ∎
We are now ready to prove Theorem 14.
If is a graph and is a binary sequence, then the chromatic number satisfies the following:
for all .
For the upper bound, note that every transitive or anti-transitive step introduces an independent set, so using a new color for each time-step, we achieve a proper coloring of with colors.
For the lower bound, first note that if is the all-1’s sequence, then by repeated application of Lemma 12, we have , so we are done. Let be the first index such that . We find that , so . It is straightforward to see that
A dominating set in a graph is a set such that . The domination number, denoted by , is the minimum size of such a set. Given two vertices we will say the closed neighborhoods of and partition the vertex set of , or simply and partition the vertex set of , if and .
We show that in ILM graphs with at least two bits equal to zero, the domination number will be at most 3 after a sufficient number of time-steps. However, for ILT graphs a straightforward discussion will show that transitive steps preserve the domination number, or more specifically for a graph we have for all
Note that if is a dominating set in , then will also dominate in , so . Assume is a dominating set of . If , then also dominates in , so . Otherwise, there exists some clone , say is the clone of some vertex , such that . Note that is a dominating set since , which implies that we can always find a dominating set of in , and thus, . By induction, this implies that , and we are done.
We have the following theorem on the domination number of general ILM graphs.
Let be a graph and be an binary sequence with at least two bits equal to . If is the second index such that , then for all ,
Let , and let be the largest and second largest indices such that . Let be any vertex in . Since , . Let be the anti-transitive clone of . and let denote the anti-transitive clone of when we perform the anti-transitive step at time . We claim that is a dominating set of .
Any vertex in that is not adjacent to is adjacent to by definition, so we need only focus on the new vertices in . Note that by the definition of an anti-transitive step. Since there are only transitive steps between time and , we also have that . Thus, any vertex is adjacent to at most one of or , and so any newly created vertex is adjacent to at least one of or . Thus, is a dominating set of . Then there are only transitive steps between time and time . The result follows from the fact that ILT steps preserve the domination number. ∎
The preceding theorem tells us that the domination number of an ILM graph after two anti-transitive steps is either , , or . After an anti-transitive step, we cannot have a dominating vertex, so that leaves the possibility of domination number or . We can characterize exactly when the domination number is and when it is . To do so, first, we establish a helpful lemma.
Let be a graph. If contains a pair of vertices whose closed neighborhoods partition the vertex set, then the same pair also partition the vertex set in and . Further, if contains a pair of vertices whose closed neighborhoods partition the vertex set, then does as well.
First, let be a pair of vertices that partition the vertex set of . In , every anti-transitive clone of the vertices in is adjacent to and each clone of a vertex in is adjacent to , so . It is straightforward to see that . so and partition the vertex set of .
Similarly, we claim that and will partition the vertex set of . Indeed, every transitive clone of a vertex in is adjacent to , and every clone of a vertex in is adjacent to , and