Complex networks are pervasive in the real world, capturing dyadic interactions between pairs of ... more Complex networks are pervasive in the real world, capturing dyadic interactions between pairs of vertices, and a large corpus has emerged on their mining and modeling. However, many phenomena are comprised of polyadic interactions between more than two vertices. Such complex hypergraphs range from emails among groups of individuals, scholarly collaboration, or joint interactions of proteins in living cells. Complex hypergraphs and their models form an emergent topic, requiring new models and techniques. A key generative principle within social and other complex networks is transitivity, where friends of friends are more likely friends. The previously proposed Iterated Local Transitivity (ILT) model incorporated transitivity as an evolutionary mechanism. The ILT model provably satisfies many observed properties of social networks, such as densification, low average distances, and high clustering coefficients. We propose a new, generative model for complex hypergraphs based on transitivity, called the Iterated Local Transitivity Hypergraph (or ILTH) model. In ILTH, we iteratively apply the principle of transitivity to form new hypergraphs. The resulting model generates hypergraphs simulating properties observed in real-world complex hypergraphs, such as densification and low average distances. We consider properties unique to hypergraphs not captured by their 2-section. We show that certain motifs, which are specified subhypergraphs of small order, have faster growth rates in ILTH hypergraphs than in random hypergraphs with the same order and expected average degree. We show that the graphs admitting a homomorphism into the 2-section of the initial hypergraph appear as induced subgraphs in the 2-section of ILTH hypergraphs. We consider new and existing hypergraph clustering coefficients, and show that these coefficients have larger values in ILTH hypergraphs than in comparable random hypergraphs.
The game of Tic-Tac-Toe is well known. In particular, in its classic version it is famous for nei... more The game of Tic-Tac-Toe is well known. In particular, in its classic version it is famous for neither player having a winning strategy. While classically it is played on a grid, it is natural to consider the effect of playing the game on richer structures, such as finite planes. Playing the game of Tic-Tac-Toe on finite affine and projective planes has been previously studied. While the second player can usually force a draw, for small orders the first player has a winning strategy. In this regard, a computer proof that Tic-Tac-Toe played on the affine plane of order 4 is a first player win has been claimed. In this note we use techniques from the theory of latin squares and transversal designs to give a human verifiable, explicit proof of this fact.
We consider social networks of competing agents that evolve dynamically over time. Such dynamic c... more We consider social networks of competing agents that evolve dynamically over time. Such dynamic competition networks are directed, where a directed edge from nodes u to v corresponds a negative social interaction. We present a novel hypothesis that serves as a predictive tool to uncover alliances and leaders within dynamic competition networks. Our focus is in the present study is to validate it on competitive networks arising from social game shows such as Survivor and Big Brother.
Complex Networks and Their Applications VIII, 2019
Competition networks are formed via adversarial interactions between actors. The Dynamic Competit... more Competition networks are formed via adversarial interactions between actors. The Dynamic Competition Hypothesis predicts that influential actors in competition networks should have a large number of common out-neighbors with many other nodes. We empirically study this idea as a centrality score and find the measure predictive of importance in several real-world networks including food webs, conflict networks, and voting data from Survivor.
We consider social networks of competing agents that evolve dynamically over time. Such dynamic c... more We consider social networks of competing agents that evolve dynamically over time. Such dynamic competition networks are directed, where a directed edge from nodes u to v corresponds a negative social interaction. We present a novel hypothesis that serves as a predictive tool to uncover alliances and leaders within dynamic competition networks. Our focus is in the present study is to validate it on competitive networks arising from social game shows such as Survivor and Big Brother.
Complex networks are pervasive in the real world, capturing dyadic interactions between pairs of ... more Complex networks are pervasive in the real world, capturing dyadic interactions between pairs of vertices, and a large corpus has emerged on their mining and modeling. However, many phenomena are comprised of polyadic interactions between more than two vertices. Such complex hypergraphs range from emails among groups of individuals, scholarly collaboration, or joint interactions of proteins in living cells. Complex hypergraphs and their models form an emergent topic, requiring new models and techniques. A key generative principle within social and other complex networks is transitivity, where friends of friends are more likely friends. The previously proposed Iterated Local Transitivity (ILT) model incorporated transitivity as an evolutionary mechanism. The ILT model provably satisfies many observed properties of social networks, such as densification, low average distances, and high clustering coefficients. We propose a new, generative model for complex hypergraphs based on transitivity, called the Iterated Local Transitivity Hypergraph (or ILTH) model. In ILTH, we iteratively apply the principle of transitivity to form new hypergraphs. The resulting model generates hypergraphs simulating properties observed in real-world complex hypergraphs, such as densification and low average distances. We consider properties unique to hypergraphs not captured by their 2-section. We show that certain motifs, which are specified subhypergraphs of small order, have faster growth rates in ILTH hypergraphs than in random hypergraphs with the same order and expected average degree. We show that the graphs admitting a homomorphism into the 2-section of the initial hypergraph appear as induced subgraphs in the 2-section of ILTH hypergraphs. We consider new and existing hypergraph clustering coefficients, and show that these coefficients have larger values in ILTH hypergraphs than in comparable random hypergraphs.
The game of Tic-Tac-Toe is well known. In particular, in its classic version it is famous for nei... more The game of Tic-Tac-Toe is well known. In particular, in its classic version it is famous for neither player having a winning strategy. While classically it is played on a grid, it is natural to consider the effect of playing the game on richer structures, such as finite planes. Playing the game of Tic-Tac-Toe on finite affine and projective planes has been previously studied. While the second player can usually force a draw, for small orders the first player has a winning strategy. In this regard, a computer proof that Tic-Tac-Toe played on the affine plane of order 4 is a first player win has been claimed. In this note we use techniques from the theory of latin squares and transversal designs to give a human verifiable, explicit proof of this fact.
We consider social networks of competing agents that evolve dynamically over time. Such dynamic c... more We consider social networks of competing agents that evolve dynamically over time. Such dynamic competition networks are directed, where a directed edge from nodes u to v corresponds a negative social interaction. We present a novel hypothesis that serves as a predictive tool to uncover alliances and leaders within dynamic competition networks. Our focus is in the present study is to validate it on competitive networks arising from social game shows such as Survivor and Big Brother.
Complex Networks and Their Applications VIII, 2019
Competition networks are formed via adversarial interactions between actors. The Dynamic Competit... more Competition networks are formed via adversarial interactions between actors. The Dynamic Competition Hypothesis predicts that influential actors in competition networks should have a large number of common out-neighbors with many other nodes. We empirically study this idea as a centrality score and find the measure predictive of importance in several real-world networks including food webs, conflict networks, and voting data from Survivor.
We consider social networks of competing agents that evolve dynamically over time. Such dynamic c... more We consider social networks of competing agents that evolve dynamically over time. Such dynamic competition networks are directed, where a directed edge from nodes u to v corresponds a negative social interaction. We present a novel hypothesis that serves as a predictive tool to uncover alliances and leaders within dynamic competition networks. Our focus is in the present study is to validate it on competitive networks arising from social game shows such as Survivor and Big Brother.
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Papers by Rehan Malik