The self-similarity of complex networks: From the view of degree–degree distance

Self-similarity of complex networks has been discovered and attracted much attention. However, the self-similarity of complex networks was measured by the classical distance of nodes. Recently, a new feature, which is named as degree–degree distance, is used to measure the distance of nodes. In the definition of degree–degree distance, the relationship between two nodes is dependent on degree of nodes. In this paper, we explore the self-similarity of complex networks from the perspective of degree–degree distance. A box-covering algorithm based on degree–degree distance is proposed to calculate the value of dimension of complex networks. Some complex networks are studied, and the results show that these networks have self-similarity from the perspective of degree–degree distance. The proposed method for measuring self-similarity of complex networks is reasonable.

Resistance distance in directed cactus graphs

Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set $V=\{1,\dotsc,n\}$. The classical distance $d_{ij}$ between any two vertices $i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$ and $j$. The resistance distance (or, simply the resistance) between any two vertices $i$ and $j$ in $V$ is defined by $r_{ij}:=l_{ii}^{\dagger}+l_{jj}^{\dagger}-2l_{ij}^{\dagger}$, where $l_{pq}^{\dagger}$ is the $(p,q)^{\rm th}$ entry of the Moore-Penrose inverse of $L$ which is the Laplacian matrix of $G$. In practice, the resistance $r_{ij}$ is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between $i$ and $j$ is always less than or equal to the classical distance, i.e., $r_{ij} \leq d_{ij}$. However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs.

Objects

Chapter One engages with Barthes’s discussion of a classical distance from worldly objects and of a modern – nouveau roman-esque – chosisme (which is predicated, Barthes says, upon a certain proximity between people and things) in order to bring the landscapes of Proust’s novel into relief. The chapter also reads Barthes’s paradoxical integration of Proustian elements within his own writing on the nouveau roman as a reflection – a rewriting – of the liminal (the simultaneously classical and modern) texture of À la recherche itself.

Proximity Between Selfadjoint Operators and Between Their Associated Random Measures

We study how the proximity between twoselfadjoint bounded operators, measured by a classical distance, canbe expressed by a proximity between the associated spectralmeasures. This last proximity is based on a partial order relation on the set of projectors. Assuming an hypothesis of commutativity, we show that the proximity between operators implies the one between theassociated spectral measures, and conversally, the proximity between spectral measures implies the one between associated selfadjoint operators.

Inhibition of Occluded Facial Regions for Distance-Based Face Recognition

This work focuses on the design and validation of a CBR system for efficient face recognition under partial occlusion conditions. The proposed CBR system is based on a classical distance-based classification method, modified to increase its robustness to partial occlusion. This is achieved by using a novel dissimilarity function which discards features coming from occluded facial regions. In addition, we explore the integration of an efficient dimensionality reduction method into the proposed framework to reduce computational cost. We present experimental results showing that the proposed CBR system outperforms classical methods of similar computational requirements in the task of face recognition under partial occlusion.

Pattern Completion in Symmetric Threshold-Linear Networks

Threshold-linear networks are a common class of firing rate models that describe recurrent interactions among neurons. Unlike their linear counterparts, these networks generically possess multiple stable fixed points (steady states), making them viable candidates for memory encoding and retrieval. In this work, we characterize stable fixed points of general threshold-linear networks with constant external drive and discover constraints on the coexistence of fixed points involving different subsets of active neurons. In the case of symmetric networks, we prove the following antichain property: if a set of neurons [Formula: see text] is the support of a stable fixed point, then no proper subset or superset of [Formula: see text] can support a stable fixed point. Symmetric threshold-linear networks thus appear to be well suited for pattern completion, since the dynamics are guaranteed not to get stuck in a subset or superset of a stored pattern. We also show that for any graph G, we can construct a network whose stable fixed points correspond precisely to the maximal cliques of G. As an application, we design network decoders for place field codes and demonstrate their efficacy for error correction and pattern completion. The proofs of our main results build on the theory of permitted sets in threshold-linear networks, including recently developed connections to classical distance geometry.